What is the difference between the Google Play store and the Amazon app store?
If you are targeting Kindle Fire tablets and the Kindle phones, you should release your app through Amazon app store because it is the default app store on those devices and will give you exposure when users search apps. If you are only targeting Android devices, Amazon app store is not a must-do. It is just another distribution channel. It has some promotional features that you may want to take advantage of such as the Free app of the day/week etc. Some white label Android tablets also uses Amazon app store as their default app store.
Google Play Store is the official app distribution method on Android devices with Google ecosystem. It is not provided for installation on devices that did not come with it pre-installed. This app store is restricted to authorized partners that have passed Google's GMS certification.
Amazon Appstore is the official app distribution method on Amazon's own devices such as Fire tablets and Fire TV. This app store is also provided by Amazon for free so that it can be installed and used on other non-Amazon devices running Android.
Both these app stores allow you to install and apps, and keep them up-to-date.
Basically Google Play is your 'default' app store, and Amazon is just a 3rd party store. Google Play can install apps automatically, and gets the benefits of Google Play Services while Amazon can't - But Amazon has the famous 'free app of the day'(Amazon Underground) - So... My opinion is to just keep both :D
Actually, the Amazon App store is NOT a third-party app store. It comes as a system app for tablets. The most common devices that the Amazon App store is installed as the main app store is the Kindle Fire devices. Either one can be a FIRST party app store but with phones, it is usually the play store.
I recommend you use the Google Play Store because there are way more apps than the Amazon App store.